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Truncated Total Least Squares: A New Regularization Method for the Solution of ECG Inverse Problems Guofa Shou, Ling Xia*, Mingfeng Jiang, Qing Wei, Feng Liu, and Stuart Crozier, Member, IEEE

Abstract—The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation . In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart–lung–torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.

=

Index Terms—ECG inverse problem, truncated total least squares (TTLS) method, h-adaptive BEM, Tikhonov, truncated single value decomposition (TSVD). Manuscript received April 17, 2007; revised September 7, 2007. This work was supported in part by the 973 National Key Basic Research & Development Program under Grant 2003CB716106, the 863 High-tech Research & Development Program under Grant 2006AA02Z307, the National Natural Science Foundation of China under Grant 30570484, the Program for New Century Excellent Talents in University under Grant NCET-04-0550 and in part by the Australian Research Council. Asterisk indicates corresponding author. G. Shou is with the Department of Biomedical Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). *L. Xia is with the Department of Biomedical Engineering, Zhejiang University, 28 Zheda Road, Hangzhou 310027, China (e-mail: [email protected]). M. Jiang is with the Department of Biomedical Engineering, Zhejiang University, Hangzhou, 310027, China, and the College of Electronics and Informatics, Zhejiang Sci-Tech University, Hangzhou 310018, China (e-mail: [email protected]). Q. Wei, F. Liu, and S. Crozier are with the School of Information Technology and Electrical Engineering, University of Queensland, Brisbane QLD 4072, Australia (e-mail: [email protected]; [email protected]; [email protected]. edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2007.912404

I. INTRODUCTION

I

N the study of inverse problems related to Electrocardiography (ECG) [1]–[3], typical target reconstruction objectives generally include: 1) equivalent sources [2] (for example, moving dipole [4], multiple dipoles [5]); 2) cardiac potential distributions (such as epicardial potentials (EPs) [3], [6]–[8], and transmembrane potentials [7], [9]); 3) cardiac activation maps/isochrones [8], [10]–[12]. For all, the main task is to interpret the electrical activities of the heart from the recorded body surface potentials (BSPs), which involves the so-called “forward” and “inverse” problems [13]. The “forward” problem deals with the modeling of the potential distribution on the body surface from equivalent cardiac sources or EPs and is an important component for the model-based “inverse” problem, in which the cardiac information are estimated based on forward solution including the transfer matrix and solution criteria. In this paper, we investigate the reconstruction of EPs from BSPs, which requires the solution of the Laplacian equation with the Cauchy boundary conditions [3], [13]: in on on

Find

on the

(1)

and are the potenwhere is the quasi-static potential, and body surface , which tials on the epicardial surface encloses the volume conductor is the tissue-dependent conand ductivity tensor. The final numerical formulation for are

(2) where is the transfer-matrix. The most widely used numerical methods for the solution of electromagnetic problems include the boundary element method (BEM) [14]–[16], the finite element method (FEM) [17]–[19], the finite difference method (FDM) [20], and the finite volume method (FVM) [21]. Of these, the BEM, an integral equation technique, is well suited for modeling surfaces; and the latter three are all differential equation techniques and are typically applied for volume conductor problems. All of these methods use meshes to model the structure; recently, a meshless method [22] was successfully employed for cardiac modeling. For heart modeling, it is reasonable to treat the conducting medium as isotropic and the conductivity values as regionally inhomogeneous (change/jump

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at organ interfaces), although the effect of the conductivity distributions to the bio-electromagnetic problems is still under active debate [22]–[25]. For the studies herein, multiple regions of homogeneous media were assumed and the BEM technique is applied to formulate the problem. To achieve high numerical accuracy for both the forward and inverse solutions, the h-adaptive BEM method is applied to the forward problem [26]. In previous work, high-order BEM [14], [16] and adaptive FEM [19] have been applied to the ECG problem, and we have employed the adaptive BEM for ECG forward modeling [26]. The adaptive BEM technique was introduced in the 1980s and has been widely used in many fields [27], [28]. Compared with traditional BEM formulation, the adaptive BEM can self-adjust the number and size of the BE meshes guided by an error indicator and thus can reduce computational time while achieving optimal BE meshes. The adaptive BEM involves three parts [27]: error estimation, adaptive tactics, and mesh refinement processes. The error estimation process evaluates the discretization errors of the boundary element solutions and the adaptive tactics select the elements to be refined and the relevant mesh refinement scheme. The mesh refinement can be further classified into h-, p-, and r-schemes and/or combinations therein. In the h-refinement scheme, the total number of elements is increased but the order of interpolation function remains invariant; in the p-scheme, the initially assumed mesh is unchanged but the order of interpolation function is increased; while in the r-scheme, both the number of elements and the order of interpolation function are kept invariant but the mesh points are relocated. In this paper, the h-refinement scheme is considered. It is well known that the transfer matrix is ill-conditioned and that the cardiac inverse problem can be characterized as a discrete, ill-posed problem [29], [30]. Small measurement errors in the body surface potentials or geometry errors in the volume conductors lead to large perturbations in the EPs. Therefore, the EPs cannot be calculated using a simple least squares method as follows:

In addition to regularization, there are other techniques available for the solution of ECG inverse problems. Temporal and spatial constraints techniques [9], [37], level-set methods [11], and statistical parameter-based methods [38] are also used for inverse problem solution; Ramanathan et al. [39] used the generalized minimal residual (GMRes) method to reconstruct EPs without adding any constraints and obtained similar solutions to those given by Tikhonov regularization. In another study, Modre et al. [12] utilized an iterative algorithm to reconstruct the myocardial activation time image. In the forward and inverse problems of ECG, errors/noise . exist on both sides of the linear system equation The vector contains the measurement errors, which arise from the recording procedure for BSPs. The transfer matrix , however, is contaminated by system noise such as model discretization error and geometric errors. In general, discretization error is introduced numerically and might be manageable; however, the geometric errors which are normally introduced during the imaging-based reconstruction of a heart-torso model, can sometimes be quite large. Both sorts of errors have been studied before [23], [25], [40]–[42]. However, most of the methods for error modeling, such as Tikhonov and TSVD, shared a common assumption, i.e., only the right-hand side of the (2) is contaminated by measurement errors, and less attention has been paid to the system errors appearing in matrix . In this paper, we apply the truncated total least squares (TTLS) method [43]–[45] to the ECG inverse problem. The TTLS method can process the measurement and geometric errors at the same time and has been successfully applied in other fields, such as ultrasound inverse scattering [46]. The proposed algorithm is tested using a realistically shaped heart–lung–torso model with inhomogeneous conductivities, and the performance is compared with other available regularization approaches.

(3)

To study the ECG inverse problem using a model-based method, the first step is to construct both a forward model and construct the transfer matrix . Here, we adapt our previously developed model obtained from CT scans of the human body [36], [47] for the current investigation. Fig. 1 shows our original heart, lung, and torso models, in which there are about 165 nodes and 315 elements in the heart, 297 nodes and 586 elements in the lung, and 210 nodes and 416 elements in the torso, respectively. Details of the model parameters can be found in [47] and references therein. In this paper, we keep the heart and lung model unmodified and employ the h-adaptive BEM technique to refine the torso model (see Fig. 2) [26]. In our new torso model, there are about 421 nodes and 838 elements, and in particular, dense meshes are used in the frontal chest area. Details of the procedure of h-adaptive BEM can be found in our previous paper [26]. Here, only the main steps are described. The h-adaptive BEM starts from the calculation of electrical potentials from the initial triangular BE meshes. Then the discretization errors of the boundary element solutions are esand the error timated by the whole meshes’ error estimation . The former parameter determines indicator of each mesh

where symbol denotes the Euclidean norm of vector space. In order to obtain a stable solution, a regularization method is typically used. For example, Tikhonov regularization with zeroth, first and second orders [1], [3], [30], [31] have been frequently used to deal with the ill-posed nature of the problem by imposing constraints on the magnitude or derivatives of the EPs. Truncated singular-value decomposition (TSVD) [1], [3], [13] is another effective method to solve the problem using a truncation technique which ignores small singular values. These two methods are called direct regularization methods, requiring an accurate choice of a regularization parameter. A variety of schemes have been developed for the determination of “smart” regularization parameters [29], including composite residual and smoothing operators (CRESO) [31], L-curve [32], discrepancy principle (DP) [33], generalized cross-validation (GCV) [34], and zero crossing methods [35]. It is recognized that the above regularization schemes are mostly problem dependent and our comparison study showed that the GCV approach is the most robust scheme [36] for our applications.

II. METHOD A. Forward Problem

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convergence. The final BE meshes are then obtained. In the procedure of h-adaptive BEM, error estimation is important, and we take the residual-type error indicator as shown in Fig. 1 in Bächtold’s paper [28] in which the error is estimated by the residual of boundary integral equation [26]. The triangular meshes are rearranged using joint quality factors of triangular pairs [48]. The refinement degree of the element is controlled by a parameter (k) in Fig. 3, which is the flowchart for the h-adaptive BEM mesh generation. B. Inverse Problem 1) The TTLS Method: The total least squares (TLS) method is a generalized version of the original least squares method, and in which both and it is motivated by linear models have errors [41]. Instead of using standard least squares formulation [29], we now state the problem with TLS formulation [41] as follows:

subject to Fig. 1. Original geometry model of heart, lung, and torso [36], [47]. Node and element information: heart (165, 315), lung (297, 586), torso (210, 416).

(4)

where denotes the Frobenius norm, and are the error and , respectively. Corresponding to TSVD versions of and Tikhonov regularizations (see the Appendix) in LS, the regularization in TLS includes the truncated TLS (TTLS) method [44] and the regularized TLS (R-TLS) [45]. In this work, we apply the TTLS method to consider both sides of the matrix with noise at the same time. The TTLS algorithm can be summarized as follows [44]. TTLS Algorithm: 1. Compute the SVD of the augmented matrix

(5) 2. Select the regularization parameter . as 3. Block-partition

(6) 4. Compute the TTLS solution

as

(7) Fig. 2. Refined geometry model using h-adaptive BEM. Node and element information: heart (165, 315), lung (297, 586), torso (421, 838). The red nodes show the electrodes applied to the ECG inverse problem.

the stopping criteria for refinement/iteration, and the latter one is used to find those elements to be refined by adding new nodes in the center of the refined triangles. The mesh refinement procedure is repeated and triangular meshes are reconstructed until

is the pseudoinverse of and . The small singular values of are neglected through selection of the regularization parameter . We note that the determination of the regularization parameter is not an easy task and after a series of investigations, we find that the generalized cross-validation (GCV) is the best approach and can lead to optimal solution for the ECG inverse problem [36]. Therefore, GCV has been our default method for the determination of the regularization parameters. In (7),

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Fig. 3. Flowchart of the construction of transfer matrix A using h-adaptive BEM approach: A: Flowchart of adaptive triangular mesh generator. B: Process of ^ : body surface potentials; R^ : error indicator of the element 0 ; R^ : error indicator of the overall element. refinement of an element. Here, 8

Fig. 4. Epicardial and body surface potentials distribution induced by a dipole setting in the heart center; the left column is the anterior side, while the right is the posterior side. (For detailed setting, see text.)

C. Evaluation of the Inverse Solution The inverse solution is evaluated using relative error (RE) and/or correlation coefficient (CC), and RE is defined as

RE

(8)

and CC is given by CC is the mean value of the EPs, where “exact” solution from simulation.

(9) is the assumed

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III. NUMERICAL SIMULATIONS In this paper, the performance of the TTLS method was tested by the reconstruction of EPs from BSPs in a realistic heart–lung–torso model. The EPs are generated from a dipole inside the heart and are normalized in the range of 10 mV [39]. Fig. 4 shows the corresponding epicardial and torso surface potentials distributions. It is noted that the visualization of the EP distributions is implemented with an extra interpolation procedure [49] on the triangulated 3-D surface of the heart model. In the inverse calculation of epicardial potentials, we choose only part of the nodes over the torso surface (248 nodes, marked with red nodes in Fig. 2), which are those measurable nodes for clinical practice. Therefore, the dimension of the transfer matrix is 248 165. The maximum and minimum singular values of are , respectively. It can be seen that the linear system is severely ill-posed, with a condition . number of To comprehensively simulate the noise involved in the inverse problem, we modify the linear system equation as

TABLE I SUMMARY OF THE INVERSE SOLUTIONS FOR THE CASE WITH MEASUREMENT NOISE IN BSPS. NOISE PROPERTY: GAUSSIAN WHITE NOISE, 50-dB AND 30-dB SNR. REGULARIZATION APPROACHES: TIKHONOV, TSVD, AND TTLS METHODS

(10) where the body surface potential is added with simulated various Gaussian white noise, and the transfer matrix is added with geometry error matrix , which is a random error generated by a Matlab function randn. This noise is normally distributed with zeroth mean value and unit standard variation, . Different error levels and normalized such that are considered by adjusting the parameter . This error setup somehow different from other approaches [3], [25], [37] such as moving the heart model inward or outward about 1 cm in the torso model, or considering the cardiac motion. In this paper, we characterized all these sorts of errors using random noise in the transfer matrix . The regularization methods, Tikhonov, TSVD, and TTLS are carried out in the Regularization Tools package under the MATLAB environment [50]. In the first simulation, only measurement noise is considered, and the noise level is about 50- and 30-dB signal-to-noise ratio (SNR) [9], [25], [37]. The matrix for Tikhonov, TSVD, and TTLS is the identity matrix, the first, and the second derivative operators, respectively. The simulation results are summarized in Table I. From this table, we can see that if there is no measurement noise in BSPs, the three regularization methods are all able to achieve good inverse solutions. With increases in noise levels, the inverse solutions are less accurate. Interestingly, the second-order regularization performs much poorer than the other two. Therefore, in the following studies, we only consider the zeroth-order regularization, which is also the most popular approach for ECG inverse problem solution [3], [8], [35]. It is also found that the three regularization approaches can lead to similar results, with Tikhonov and TTLS performing slightly better than TSVD. In the second simulation study, geometric noise is considered, and the control parameter is adjusted in a range of 0.01–0.1. Table II summarizes the inverse solution results from three zeroth-order regularization methods with both measurement and

TABLE II SUMMARY OF THE INVERSE SOLUTIONS FOR THE CASE WITH MEASUREMENT NOISE IN BSPS AND GEOMETRY ERRORS IN TRANSFER MATRIX A. NOISE PROPERTY: MEASUREMENT NOISE: GAUSSIAN WHITE NOISE, 50-dB AND 30-dB SNR; GEOMETRY NOISE: RANDOM ERROR. REGULARIZATION APPROACHES: TIKHONOV, TSVD, AND TTLS METHODS (L: IDENTITY MATRIX)

geometric noise. In this case, TTLS method performs much better than Tikhonov and TSVD, especially for the case with geometric errors only. To confirm TTLS’s performance, we consider the geometry is error in a relatively larger range, where the parameter within the range of 0.001–1. The corresponding RE and CC

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Fig. 5. Inverse solution for the case geometry error involved only. Regularization solver: Tikhonov, TSVD, and TTLS.

values are shown in Fig. 5(A) and (B). From this test, we observe that the TTLS method produces the better inverse soluis equal to 0.01 and 0.1, the RE value of TTLS tion. When method is decreased by about 10%, and CC value is increased by 5% compared with those of the Tikhonov and the TSVD methods. Fig. 6 demonstrates the EP distribution reconstructed . It is clear that the EP distrifrom the case where bution generated by TTLS is closer to the forward model result (see Fig. 4), particularly for the maximum and minimum value locations. In addition, the TTLS reported epicardial potential profile is smoother than those of Tikhonov and TSVD methods. In this paper, the TTLS method is implemented in Matlab, and it only takes about 2 s for a solution on a Dell computer with Intel Core2 CPU 1.80 GHz, and 2.00-GB RAM. The algorithm is therefore efficient and can be used for real-time applications. IV. DISCUSSION In the ECG inverse problem, in addition to the measurement errors in the BSPs, the transfer matrix is also contaminated by noise which is contributed to by: geometric inaccuracies in the model anatomy, the beating of the heart, and the representation of conductivities of the volume conductor. These system errors make the estimation problem very difficult, and it is challenging to find a good, stable inverse solution. In this paper, we use the TTLS regularization method as introduced by Fierro et al. [44] to solve this discrete ill-posed inverse problem of ECG. The TTLS method was implemented and compared with the most commonly used regularization methods, including Tikhonov and TSVD. These three regularization approaches achieve similar solutions when there is only measurement error. However, TTLS is able to produce much better results than Tikhonov and TSVD when there are geometric errors involved. The results from the TTLS method, when compared to those of Tikhonov and TSVD methods can be readily understood by examining the geometric meaning of the TLS [43]: Consider a curve that minimizes the distances from the curve to a given set of points, then it is the vertical distance used in the LS approach, while the perpendicular distance is used in the TLS approach. When there is only measurement error, the curve is substantially flat, and the vertical and perpendicular distances are equivalent, leading to inverse solutions at the same level of accuracy. If there is error in , however, the curve’s slope is bigger, and the perpendicular distance is shorter than vertical

distance, which makes the TLS solution better than the LS. The results also support that the TTLS method suits strongly ill-posed problems, which has been pointed out by Fierro et al. [44]. The TTLS performance is evident by the RE, CC values and EP profiles obtained in the simulations in this ECG inverse problem study. The TTLS algorithm is capable of handling both geometry errors and measurement noise, which is important for practical applications. To achieve more accurate inverse solutions in the presence of noise, careful solution constraints, such as additional time and spatial constraints, need further investigation. Another feature of this study is that we have applied the h-adaptive BEM for the ECG forward modeling. The h-adaptive BEM uses an iteration procedure to find the optimal BE meshes for torso representation. The algorithm has been detailed in our previous paper [26], in which both a concentric spheres model and a realistic heart–lung–torso model are investigated. Through forward and inverse ECG modeling, the h-adaptive BEM has been validated and is demonstrated to be able to improve the numerical accuracy of the BEM with minimal computational cost. Therefore, the h-adaptive BEM is a promising technique to construct the forward model for inverse studies. The current torso–lung–heart model is reconstructed from the CT scans of the human body; MRI-based solution may be better for the reconstruction of the model, especially for the consideration of cardiac motion. The TTLS seems more robust than other regularization schemes when the model geometry becomes an important issue in potential clinical applications. In addition, our adaptive BEM scheme can represent the model geometry well and therefore further improve the system accuracy. Further investigations are underway toward practical usage. V. CONCLUSION In this paper, the TTLS method is proposed to study inverse problems in ECG, with the inclusion of noise on both sides of the linear system equation. The simulations are implemented in a realistic heart–lung–torso model constructed using a h-adaptive BEM technique. The results demonstrate that the proposed method can produce better solutions than conventional regularization methods and that the zeroth-order regularization is optimal for the ECG inverse problem. In this paper, the reconstruction of EPs from BSPs only has been considered; in future work,

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Fig. 6. Epicardial potential distributions reconstructed from body surface potentials. Noise property: geometry error only,  = 0:01. Left panel: anterior view; right panel: posterior view. (A) Zeroth-order Tikhonov method; (B) TSVD method; and (C) TTLS method.

we will employ this promising scheme to other cardiac source reconstruction and imaging problems.

The ordinary LS solution can be written as

(12) APPENDIX According to matrix theory, the singular value decomposition (SVD) of is given by

(11) where thonormal columns, .

and

have orwith

Due to the division by small singular values , the solution may be dominated by components associated with the . Therefore, regularization is necessary to stabilize errors in the solution. The common and well-known regularization approach is Tikhonov regularization [36], which approximately solves the by minimizing linear system (13) where is a positive regularization parameter chosen to control the size of the solution vector and is a matrix that defines

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a (semi)norm on the solution through which the “size” is measured. In general, represents the first or second derivative operator (first-order or second-order). The equivalent problem of (13) is (14) As increases, the (semi)norm of the solution vector decreases monotonically while the residual increases monotonically. It is easy to prove that if , then the solution of (14) is given by

(15) Therefore, Tikhonov regularization suppresses (filters) the components of the solution corresponding to the small singular values of . The regularization in TSVD method is achieved by truncating , where is the equivalent regularthe singular value for ization parameter

(16)

ACKNOWLEDGMENT The authors would like to thank the three reviewers for helpful comments on the paper. REFERENCES [1] O. Dössel, “Inverse problem of electro- and magnetocardiography: Review and recent progress,” Int. J. Bioelectromag., vol. 2, no. 2, 2000. [2] R. M. Gulrajani, P. Savard, and F. A. Roberge, “The inverse problem in electrocadiography: Solutions in terms of equivalent sources,” CRC Crit. Rev. Biomed. Eng., vol. 16, pp. 171–214, 1988. [3] Y. Rudy and B. J. Messinger-Rapport, “The inverse problem in electrocardiography: Solutions in terms of epicardial potentials,” CRC Crit. Rev. Biomed. Eng., vol. 16, pp. 215–268, 1988. [4] Y. Fukuoka, T. F. Oostendorp, D. A. Sherman, and A. A. Armoundas, “Applicability of the single equivalent moving dipole model in an infinite homogeneous medium to identify cardiac electrical sources: A computer simulation study in a realistic anatomic geometry torso model,” IEEE Trans. Biomed. Eng., vol. 53, no. 12, pp. 2436–2444, Dec. 2006. [5] D. G. Beenter and R. M. Arthur, “Estimation of heart-surface potentials using regularized multipole sources,” IEEE Trans. Biomed. Eng., vol. 51, no. 8, pp. 1366–1373, Aug. 2004. [6] C. Ramanathan, R. N. Ghanem, P. Jia, K. Ryu, and Y. Rudy, “Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia,” Nature Med., vol. 10, pp. 422–428, 2004. [7] B. Messnarz, M. Seger, R. Modre, G. Fischer, F. Hanser, and B. Tilg, “A comparison of noninvasive reconstruction of epicardial versus transmembrane potentials in consideration of the null space,” IEEE Trans. Biomed. Eng., vol. 51, no. 9, pp. 1609–2004, Sep. 2004. [8] L. K. Cheng, J. M. Bodley, and A. J. Pullan, “Comparison of potential and activation-based formulations for the inverse problem of electrocardiology,” IEEE Trans. Biomed. Eng., vol. 50, no. 1, pp. 11–22, Jan. 2003. [9] B. Messnarz, B. Tilg, R. Modre, G. Fischer, and F. Hansen, “A new spatiotemporal regularization approach for reconstruction of cardiac transmembrane potential patterns,” IEEE Trans. Biomed. Eng., vol. 51, no. 2, pp. 273–281, Feb. 2004.

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Guofa Shou was born in Zhejiang Province, China, in April 1982. He received the B.S. degree in biomedical engineering from Northwestern Polytechnical University, China. He is currently working toward the Ph.D. degree in the Department of biomedical engineering at Zhejiang University. His research interest includes the forward and inverse problem of ECG, and the numeric algorithms in bio-electromagnetism.

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Ling Xia was born in Zhejiang Province, China, in October 1965. He received the B.S. degree in electrical automation control and the Ph.D. degree in biomedical engineering from Zhejiang University, China, in 1987 and 1996, respectively. He is currently a Professor and the Vice Director of the Institute of Biomedical Engineering of Zhejiang University. His research interest includes the multiscale heart modeling and simulation, the ECG inverse problem, MRI key technology, and the biological effect of electromagnetic field. He has published more than 40 papers in journals and obtained several highest-level grants from Chinese government.

Mingfeng Jiang received the B.S. degree and the M.S. degree in biomedical engineering from Chongqing University, China, in 2000 and 2003, respectively. He is currently working toward the Ph.D. degree in the Department of biomedical engineering in Zhejiang University, China. His current research interests are in the forward and inverse problem of ECG and in the biomedical signal processing.

Qing Wei received the M.S. degree from University of Queensland (UQ), Brisbane, Australia, in 2002. She is currently working toward the Ph.D. degree in the school of information technology and electrical engineering, UQ. Her current research interests include numerical tool design in biomedical engineering and scientific visualization.

Feng Liu received the Ph.D. degree in biomedical engineering from Zhejiang University, Hangzhou, China, in 2000. He is currently a Senior Research Fellow at the School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, Australia. His current research interests include magnetic resonance engineering, bioelectromagnetism, and power electronics.

Stuart Crozier (M’93) received the Ph.D. degree in electrical engineering and the D.Eng. degree in biomedical engineering from the University of Queensland (UQ), Brisbane, Australia, in 1991 and 2002, respectively. He is currently a Research Director in the School of Information Technology and Electrical Engineering, UQ. He is the author or coauthor of more than 100 journal articles and is the holder of numerous patents. His current research interests include magnetic resonance engineering, bioelectromagnetics, and the methodological development of magnetic resonance. Prof. Crozier is an Associate Editor of the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING.